Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2013, Article ID 610186, 14 pages
http://dx.doi.org/10.1155/2013/610186
Research Article
Parametric Analysis of Flexible Logic Control Model
Lihua Fu, Dan Wang, and Jinyun Kuang
College of Computer Science, Beijing University of Technology, Beijing 100124, China
Correspondence should be addressed to Lihua Fu; fulihuapaper@sohu.com
Received 20 September 2012; Revised 26 December 2012; Accepted 1 January 2013
Academic Editor: Xiang Li
Copyright © 2013 Lihua Fu et al. is is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Based on deep analysis about the essential relation between two input variables of normal two-dimensional fuzzy controller, we
used universal combinatorial operation model to describe the logic relationship and gave a �exible logic control method to realize
the effective control for complex system. In practical control application, how to determine the general correlation coefficient of
�exible logic control model is a problem for further studies. First, the conventional universal combinatorial operation model has
been limited in the interval [0, 1]. Consequently, this paper studies a kind of universal combinatorial operation model based on
the interval [𝑎𝑎𝑎 𝑎𝑎𝑎. And some important theorems are given and proved, which provide a foundation for the �exible logic control
method. For dealing reasonably with the complex relations of every factor in complex system, a kind of universal combinatorial
operation model with unequal weights is put forward. en, this paper has carried out the parametric analysis of �exible logic
control model. And some research results have been given, which have important directive to determine the values of the general
correlation coefficients in practical control application.
1. Introduction
Fuzzy control has made the rapid development, and it has
found a considerable number of successful industrial applications in recent years [1]. However, from the mathematical
viewpoint, Professor Li revealed the interpolation mechanism of fuzzy control and proved that fuzzy controller is in
essence an interpolator [2]. So, there are two problems in controlling some practical complex systems. One is that control
rules will grow exponentially with the growing of inputs, and
the other one is that the precision of control system is not high
[3].
Compound controllers combine fuzzy control and other
relatively mature control methods to obtain effective control
effect, such as Fuzzy-PID controllers [4], fuzzy prediction
control [5], and adaptive fuzzy 𝐻𝐻-in�nity control [6]. To
reduce dimensionality, hierarchical fuzzy logic controller
separates the set of control rules into several sets based on different functions [7, 8]. e basic idea of adaptive fuzzy controllers based on variable universe is to keep the control rules
unchanged and change the region bound of fuzzy variables
with the values of input fuzzy variables in order to increase
control rules indirectly [9]. ough a great deal of research
has been done to improve the performance of fuzzy control,
most of these methods are based on the basic idea that fuzzy
controller is a piecewise approximation. However, to date,
there has been relatively little research conducted on the
internal relations among input variables of fuzzy controllers.
Universal Logic [10], proposed by He et al., is a kind of
�exible logic. It considers the continuous change of not only
the truth value of proposition, which is called truth value
�exibility, but also the relation between propositions, which
is called relational �exibility. Based on fuzzy logic, it puts up
two important coefficients: generalized correlation coefficient
“ℎ” and generalized self-correlation coefficient “𝑘𝑘.” e �exible
change of universal logic operations is based on “ℎ” and “𝑘𝑘.”
So, Universal Logic provides a new theoretical foundation to
realize more accurate control for complex systems.
In our previous work [3], we focused on the basic physical meanings of fuzzy input variables of the normal twodimensional fuzzy controller, such as 𝐸𝐸 and 𝐸𝐸𝐸𝐸. And we proved that the essential relation between them is just universal
combinatorial relation in Universal Logic [10]. So, the simple
universal combinatorial operation can be used instead of
complex fuzzy reasoning process. As a result, a �exible logic
control method was put forward.
rough the previous analysis, it is clear that �exible
ability of �exible logic control model is resulted from the
2
Discrete Dynamics in Nature and Society
following aspect. We use universal combinatorial operation
to re�ect the essential relation between the deviation and
the deviation change of control system, which considers
the continuous change of the relation between things. And
universal combinatorial operation model is not a single �xed
operator, but a continuous cluster of combinatorial operators
determined by the general correlation coefficient ℎ between
propositions. In practical control application, according to
the general correlation between propositions, we can take the
corresponding one from the cluster to realize effective control
for complex system.
However, in practical control application, how to determine the general correlation coefficient ℎ of �exible logic control model is a problem for further studies. First, the conventional universal combinatorial operation model has been
limited in the interval [0, 1]. Consequently, this paper studies
a kind of universal combinatorial operation model based
on the interval [𝑎𝑎𝑎 𝑎𝑎𝑎. And some important theorems are
given and proved, which provide a foundation for the control
application of universal combinatorial operation. en, this
paper carries out the parametric analysis of �exible logic
control model. And some research results have been given,
which have important directive to determine the values
of the general correlation coefficients in practical control
application.
e rest of the paper is organized as follows. Section 2
introduces necessary background on universal combinatorial
operation model and �exible logic control method and gives
and proves some important theorems of universal combinatorial operation model based on the interval [𝑎𝑎𝑎 𝑎𝑎𝑎. Section 3
carries out the parametric analysis of �exible logic control
model and gives some research results. Finally, concluding
remarks are given in Section 4.
2. Universal Combinatorial Operation Model
and Flexible Logic Control Method
2.1. Universal Combinatorial Operation Model. It is well
known that there is complex relation between every factor
of complex system, which may be con�ictive or consistent.
For dealing reasonably with the complex relations, various
aggregation operators have been given which are mostly 𝑇𝑇norm, 𝑆𝑆-norm, or Mean operators. Nevertheless, 𝑇𝑇-norm, 𝑆𝑆norm, and Mean operators have the following properties:
0 ≤ 𝑇𝑇 󶀡󶀡𝑥𝑥𝑥 𝑥𝑥󶀱󶀱 ≤ min 󶀡󶀡𝑥𝑥𝑥 𝑥𝑥󶀱󶀱
max 󶀡󶀡𝑥𝑥𝑥 𝑥𝑥󶀱󶀱 ≤ 𝑆𝑆 󶀡󶀡𝑥𝑥𝑥 𝑥𝑥󶀱󶀱 ≤ 1
min 󶀡󶀡𝑥𝑥𝑥 𝑥𝑥󶀱󶀱 ≤ 𝑀𝑀 󶀡󶀡𝑥𝑥𝑥 𝑥𝑥󶀱󶀱 ≤ max 󶀡󶀡𝑥𝑥𝑥 𝑥𝑥󶀱󶀱 .
(1)
As a result, 𝑇𝑇-norm or 𝑆𝑆-norm can only handle mutually
con�ictive relation. In contrast, Mean operators can only
handle mutually consistent relation. erefore, the operating
regions of these aggregation operators are localized. Universal combinatorial operation model is the combinatorial
connective of Universal Logic, whose operating region is the
standard interval [0, 1].
In the paper, we will only consider the generalized correlation coefficient ℎ. So, zero-order universal combinatorial
operation model is de�ned as follows.
�e�nition 1 (see [10]). Zero-order universal combinatorial
operation model is the cluster
𝐶𝐶𝑒𝑒 󶀡󶀡𝑥𝑥𝑥 𝑥𝑥𝑥𝑥󶀱󶀱
1/𝑚𝑚
= ite 󶁃󶁃Γ𝑒𝑒 󶁣󶁣󶁣󶁣𝑥𝑥𝑚𝑚 + 𝑦𝑦𝑚𝑚 − 𝑒𝑒𝑚𝑚 󶀱󶀱
󶁳󶁳 ∣ 𝑥𝑥 𝑥 𝑥𝑥 𝑥 𝑥𝑥𝑥𝑥
𝑚𝑚
1 − 󶀢󶀢Γ1−𝑒𝑒 󶁢󶁢󶁢1 − 𝑥𝑥)𝑚𝑚 + 󶀡󶀡1 − 𝑦𝑦󶀱󶀱
−(1 − 𝑒𝑒)𝑚𝑚 󶁲󶁲󶀲󶀲
1/𝑚𝑚
(2)
∣ 𝑥𝑥 𝑥 𝑥𝑥 𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥󶁔󶁔 ,
where 𝑚𝑚 𝑚𝑚𝑚 𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚 𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚 𝑚 𝑚𝑚𝑚 and 𝑒𝑒 𝑒
[0, 1].
Remark 2. e conditional expression ite{𝛽𝛽 𝛽 𝛽𝛽𝛽 𝛽𝛽𝛽 represents
that if 𝛼𝛼 is true, then the result is 𝛽𝛽, otherwise 𝛾𝛾. Similarly,
ite{𝛽𝛽1 ∣ 𝛼𝛼1 ; 𝛽𝛽2 ∣ 𝛼𝛼2 ;𝛾𝛾𝛾𝛾 ite{𝛽𝛽1 ∣ 𝛼𝛼1 ; ite{𝛽𝛽2 ∣
𝛼𝛼2 ;𝛾𝛾𝛾𝛾. And Γ1 [𝑥𝑥𝑥𝑥 ite{1 ∣ 𝑥𝑥 𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥
0 or 𝑥𝑥 is an imaginary number; 𝑥𝑥𝑥.
Universal combinatorial operation model is a continuous
cluster of combinatorial operators, which can be continuously changeable with generalized correlation coefficient ℎ
between propositions. In practical application, according
to the general correlation between propositions, we can
take the corresponding one from the cluster. As generalized
correlation coefficient h is equal to some special values, the
corresponding combinatorial operators are given in Table 1.
2.2. Universal Combinatorial Operation Model on Any Interval
[𝑎𝑎𝑎 𝑎𝑎𝑎. In practical control application, fuzzy domain of fuzzy
variables, 𝐸𝐸 and 𝐸𝐸𝐸𝐸, is mostly symmetrical, such as [−6, 6].
However, the conventional universal combinatorial operation
model has been limited in the interval [0, 1]. So, Chen [11] put
forward a kind of universal combinatorial operation model,
which is on any interval [𝑎𝑎𝑎 𝑎𝑎𝑎.
�e�nition � (see [11]). Normal universal Not operation
model in any interval [𝑎𝑎𝑎 𝑎𝑎𝑎 is de�ned as follows:
GN (𝑥𝑥) = 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
(3)
GN (𝑥𝑥) ∈ [𝑎𝑎𝑎 𝑎𝑎] ,
(4)
For the previous de�nition, some common properties of
normal universal Not operation model are given:
(1) closure
(2) two polar law
GN (𝑎𝑎) = 𝑏𝑏𝑏
(3) symmetric involution
GN (𝑏𝑏) = 𝑎𝑎𝑎
GN (GN (𝑥𝑥)) = 𝑥𝑥𝑥
(5)
(6)
Discrete Dynamics in Nature and Society
3
T 1: Some special combinatorial operators of universal combinatorial operation model.
General correlation
between propositions
Value of ℎ
1
Combinatorial operator
𝐶𝐶 (𝑥𝑥, 𝑦𝑦, 1)
= ite{min(𝑥𝑥, 𝑦𝑦) ∣ 𝑥𝑥 𝑥 𝑥𝑥 𝑥 𝑥𝑥𝑥;
max(𝑥𝑥, 𝑦𝑦) ∣ 𝑥𝑥 𝑥 𝑥𝑥 𝑥 𝑥𝑥𝑥;
𝑒𝑒𝑒
Max-attracting
0.75
Independent correlation
0.5
Max-rejecting
0
𝐶𝐶𝑒𝑒 (x, y, 0.75)
= ite{𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 𝑥𝑥 𝑥 𝑥𝑥 𝑥 𝑥𝑥𝑥;
(𝑥𝑥 𝑥 𝑥𝑥 𝑥 𝑥𝑥𝑥𝑥 𝑥 𝑥𝑥)/(1 − 𝑒𝑒)∣ 𝑥𝑥 𝑥 𝑥𝑥 𝑥 𝑥𝑥𝑥;
𝑒𝑒𝑒
𝐶𝐶𝑒𝑒 (𝑥𝑥, 𝑦𝑦, 0.5) = Γ1 [𝑥𝑥 𝑥 𝑥𝑥 𝑥 𝑥𝑥𝑥
𝐶𝐶𝑒𝑒 (x, y, 0)
= ite{0 ∣ 𝑥𝑥, 𝑦𝑦 𝑦𝑦𝑦;
1 ∣ 𝑥𝑥, 𝑦𝑦 𝑦𝑦𝑦;
𝑒𝑒𝑒
Max-restraining
�e�nition � (see [11]). Universal combinatorial operation
model on any interval [𝑎𝑎𝑎 𝑎𝑎𝑎 is the cluster
GC𝑒𝑒 󶀡󶀡𝑥𝑥𝑥𝑥𝑥𝑥𝑥󶀱󶀱
= ite 󶁆󶁆min 󶀦󶀦𝑒𝑒𝑒 (𝑏𝑏𝑏𝑏𝑏)
× 󶁦󶁦max 󶀦󶀦0,
𝑏𝑏𝑏𝑏𝑏
𝑚𝑚
𝑚𝑚
Probability combinatorial operator 𝐶𝐶𝑒𝑒2
Bounded combinatorial operator 𝐶𝐶𝑒𝑒1
−1󶀷󶀷 ∣ 𝑥𝑥 𝑥 𝑥𝑥 𝑥 𝑥𝑥
− min 󶀧󶀧0,2󶁥󶁥max 󶀥󶀥0,
(1 − 𝑥𝑥𝑥𝑚𝑚 + (1 − 𝑦𝑦𝑦𝑚𝑚 − 1
󶀵󶀵󶁵󶁵
2𝑚𝑚
1/𝑚𝑚
−1󶀷󶀷 ∣ 𝑥𝑥 𝑥 𝑥𝑥 𝑥 𝑥𝑥 𝑥󶁗󶁗 .
(8)
According to the de�nition of Universal Combinatorial
Operation Model in any interval, the following characters [11]
are attained:
+𝑎𝑎󶀶󶀶 ∣ 𝑥𝑥 𝑥 𝑥𝑥 𝑥 𝑥𝑥𝑥𝑥
× 󶁦󶁦max 󶀦󶀦0,
Zadeh combinatorial operator 𝐶𝐶𝑒𝑒3
Drastic combinatorial operator 𝐶𝐶𝑒𝑒0
1/𝑚𝑚
𝑚𝑚
(𝑥𝑥𝑥𝑥𝑥) + 󶀡󶀡𝑦𝑦𝑦𝑦𝑦󶀱󶀱 −(𝑒𝑒𝑒𝑒𝑒)
󶀶󶀶󶀶󶀶
(𝑏𝑏𝑏𝑏𝑏)𝑚𝑚
− min 󶀦󶀦𝑒𝑒′ , (𝑏𝑏𝑏𝑏𝑏)
Name of combinatorial operator
𝑒𝑒
(1) GC𝑒𝑒 (𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 conforms to the combination axiom:
(a) boundary condition GC1
𝑚𝑚
𝑚𝑚
𝑚𝑚
(𝑏𝑏𝑏𝑏𝑏) +󶀡󶀡𝑏𝑏𝑏𝑏𝑏󶀱󶀱 −(𝑏𝑏𝑏𝑏𝑏)
󶀶󶀶󶀶󶀶
(𝑏𝑏𝑏𝑏𝑏)𝑚𝑚
+𝑎𝑎󶀶󶀶 ∣ 𝑥𝑥 𝑥 𝑥𝑥 𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥󶁖󶁖 ,
(7)
where 𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚 𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚, ℎ ∈ [0, 1], 𝑚𝑚 𝑚 𝑚𝑚, 𝑒𝑒𝑒 𝑒𝑒′ ∈
[𝑎𝑎𝑎 𝑎𝑎𝑎, and 𝑒𝑒′ = GN(𝑒𝑒𝑒.
Without loss of generality, we assume that the fuzzy
domain of fuzzy variables, E and EC, is the interval [−1, 1].
As a result, when 𝑎𝑎, 𝑏𝑏, and 𝑒𝑒 are equal to −1, 1, and 0, respectively, the corresponding zero-order universal combinatorial
operator is the cluster
GC0 󶀡󶀡𝑥𝑥𝑥𝑥𝑥𝑥𝑥󶀱󶀱
when 𝑥𝑥𝑥𝑥𝑥 𝑥 𝑥𝑥, GC𝑒𝑒 (𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 𝑥 𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥;
when 𝑥𝑥𝑥𝑥𝑥 𝑥 𝑥𝑥, GC𝑒𝑒 (𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 𝑥 𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥;
when 𝑥𝑥 𝑥 𝑥𝑥𝑥 𝑥𝑥𝑥, GC𝑒𝑒 (𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥;
otherwise, min(𝑥𝑥𝑥𝑥𝑥𝑥 𝑥 GC𝑒𝑒 (𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 𝑥
max(𝑥𝑥𝑥𝑥𝑥𝑥,
1/𝑚𝑚
(b) monotonicity GC2
GC𝑒𝑒 (𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 increases monotonously along
with 𝑥𝑥 and 𝑦𝑦,
(c) continuity GC3
when ℎ ∈ (0, 1), GC𝑒𝑒 (𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 is continuous
for all 𝑥𝑥 and 𝑦𝑦,
(d) commutative law GC4
GC𝑒𝑒 󶀡󶀡𝑥𝑥𝑥𝑥𝑥𝑥𝑥󶀱󶀱 = GC𝑒𝑒 󶀡󶀡𝑦𝑦𝑦𝑦𝑦𝑦𝑦󶀱󶀱 ,
(9)
GC𝑒𝑒 (𝑥𝑥𝑥𝑥𝑥𝑥𝑥) = 𝑥𝑥𝑥
(10)
GC𝑒𝑒 󶀡󶀡𝑥𝑥𝑥𝑥𝑥𝑥𝑥󶀱󶀱 ∈ [𝑎𝑎𝑎 𝑎𝑎] ,
(11)
(e) law of identical element GC5
= ite 󶁇󶁇min 󶀧󶀧0,2
𝑚𝑚
1/𝑚𝑚
(𝑥𝑥 𝑥 𝑥)𝑚𝑚 + 󶀡󶀡𝑦𝑦 𝑦𝑦󶀱󶀱 − 1
× 󶁦󶁦max 󶀦󶀦0,
󶀶󶀶󶁶󶁶
2𝑚𝑚
(2) closure
4
Discrete Dynamics in Nature and Society
(3) inverse law
Substituting (16), (17), and (18) separately into (15),
GC𝑒𝑒 (𝑥𝑥𝑥 𝑥𝑥𝑥 𝑥 𝑥𝑥𝑥 𝑥) = 𝑒𝑒𝑒
(12)
GC𝑒𝑒 (𝑒𝑒𝑒 𝑒𝑒𝑒𝑒) = 𝑒𝑒𝑒
(13)
(4) renunciation law
GCGN(𝑒𝑒𝑒 󶀡󶀡GN (𝑥𝑥) , GN 󶀡󶀡𝑦𝑦󶀱󶀱 ,ℎ󶀱󶀱
= 𝑏𝑏𝑏𝑏𝑏𝑏 𝑏𝑏𝑏 󶀪󶀪𝑒𝑒𝑒 (𝑏𝑏𝑏𝑏𝑏)
(𝑥𝑥 𝑥 𝑥𝑥)𝑚𝑚 + 󶀡󶀡𝑦𝑦𝑦𝑦𝑦󶀱󶀱
×󶀄󶀄max 󶀧󶀧0,
(𝑏𝑏𝑏𝑏𝑏)𝑚𝑚
󶀜󶀜
eorem 5. Let GN(𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 (𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺
𝐺𝐺𝐺𝐺𝑒𝑒 (𝑥𝑥𝑥 𝑥𝑥𝑥 𝑥𝑥, 𝑥𝑥𝑥 𝑥𝑥 𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥, 𝑒𝑒 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒.
Proof. 𝑥𝑥𝑥 𝑥𝑥 𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥, 𝑒𝑒 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒, according to the closure of central generalized negation operation and universal combinatorial operation:
GN(𝑥𝑥𝑥𝑥 GN(𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦, GN(𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 and
GN(𝑒𝑒𝑒
GC
(GN(𝑥𝑥𝑥𝑥 GN(𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦, and according to the
de�nition of universal combinatorial operation:
(1) when 𝑥𝑥 𝑥 𝑥𝑥 𝑥 𝑥𝑥𝑥,
(𝑒𝑒 𝑒𝑒𝑒)𝑚𝑚 󶀅󶀅
−
󶀷󶀷
(𝑏𝑏𝑏𝑏𝑏)𝑚𝑚
󶀝󶀝
and then
= (𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏) + 󶀡󶀡𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏󶀱󶀱
(14)
= 2 (𝑏𝑏𝑏𝑏𝑏) − 󶀡󶀡𝑥𝑥 𝑥 𝑥𝑥󶀱󶀱
−min 󶀨󶀨𝑒𝑒𝑒 (𝑏𝑏𝑏𝑏𝑏)
× 󶁦󶁦max 󶀦󶀦0,
1/𝑚𝑚
= min 󶀨󶀨𝑒𝑒𝑒 (𝑏𝑏𝑏𝑏𝑏)
= 𝑏𝑏𝑏𝑏𝑏𝑏 𝑏𝑏𝑏 󶀧󶀧𝑏𝑏𝑏𝑏𝑏𝑏 GN (𝑒𝑒) , (𝑏𝑏𝑏𝑏𝑏)
(𝑥𝑥 𝑥 𝑥𝑥)𝑚𝑚
× 󶀄󶀄max 󶀨󶀨0,
(𝑏𝑏𝑏𝑏𝑏)𝑚𝑚
󶀜󶀜
(𝑏𝑏𝑏 GN (𝑥𝑥))𝑚𝑚
× 󶁥󶁥max 󶀥󶀥0,
(𝑏𝑏𝑏𝑏𝑏)𝑚𝑚
𝑚𝑚
(𝑏𝑏𝑏 GN (𝑒𝑒))𝑚𝑚
󶀵󶀵󶁵󶁵
(𝑏𝑏𝑏𝑏𝑏)𝑚𝑚
1/𝑚𝑚
+
+𝑎𝑎󶀷󶀷 .
(15)
�ccording to the de�nition of central generalized negation operation,
GN (𝑒𝑒) = 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
𝑚𝑚
󶀡󶀡𝑦𝑦𝑦𝑦𝑦󶀱󶀱 − (𝑒𝑒 𝑒𝑒𝑒)𝑚𝑚
󶀶󶀶󶀶󶀶
(𝑏𝑏𝑏𝑏𝑏)𝑚𝑚
+𝑎𝑎󶀶󶀶󶀶󶀶
GCGN(𝑒𝑒𝑒 󶀡󶀡GN (𝑥𝑥) , GN 󶀡󶀡𝑦𝑦󶀱󶀱 ,ℎ󶀱󶀱
GN 󶀡󶀡𝑦𝑦󶀱󶀱 = 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
(𝑥𝑥 𝑥 𝑥𝑥)𝑚𝑚
(𝑏𝑏𝑏𝑏𝑏)𝑚𝑚
+
�en, according to the de�nition of GC𝑒𝑒 (𝑥𝑥𝑥 𝑥𝑥𝑥 𝑥𝑥,
GN (𝑥𝑥) = 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
(19)
GN 󶀢󶀢GCGN(𝑒𝑒𝑒 󶀡󶀡GN (𝑥𝑥) , GN 󶀡󶀡𝑦𝑦󶀱󶀱 ,ℎ󶀱󶀱󶀱󶀱
> 2 (𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏) = 2GN (𝑒𝑒) .
−
+𝑎𝑎󶀺󶀺 ,
= 𝑏𝑏𝑏𝑏𝑏𝑏 󶀨󶀨𝑏𝑏𝑏𝑏𝑏
GN (𝑥𝑥) + GN 󶀡󶀡𝑦𝑦󶀱󶀱
󶀡󶀡𝑏𝑏𝑏 GN 󶀡󶀡𝑦𝑦󶀱󶀱󶀱󶀱
+
(𝑏𝑏𝑏𝑏𝑏)𝑚𝑚
1/𝑚𝑚
𝑚𝑚
(16)
(17)
(18)
= GC𝑒𝑒 󶀡󶀡𝑥𝑥𝑥 𝑥𝑥𝑥 𝑥󶀱󶀱 .
𝑚𝑚
󶀡󶀡𝑦𝑦𝑦𝑦𝑦󶀱󶀱 − (𝑒𝑒 𝑒𝑒𝑒)𝑚𝑚 󶀅󶀅
󶀸󶀸
(𝑏𝑏𝑏𝑏𝑏)𝑚𝑚
󶀝󶀝
1/𝑚𝑚
+𝑎𝑎󶀺󶀺 .
(20)
(2) When 𝑥𝑥 𝑥 𝑥𝑥𝑥 𝑥𝑥𝑥,
GN (𝑥𝑥) + GN 󶀡󶀡𝑦𝑦󶀱󶀱
= (𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏) + 󶀡󶀡𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏󶀱󶀱
= 2 (𝑏𝑏𝑏𝑏𝑏) − 󶀡󶀡𝑥𝑥 𝑥 𝑥𝑥󶀱󶀱
<2 (𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏) = 2GN (𝑒𝑒) .
(21)
Discrete Dynamics in Nature and Society
5
So, according to the de�nition of GC𝑒𝑒 (𝑥𝑥𝑥 𝑥𝑥𝑥 𝑥𝑥,
According to the de�nition of GC𝑒𝑒 (𝑥𝑥𝑥 𝑥𝑥𝑥 𝑥𝑥,
GCGN(𝑒𝑒𝑒 󶀡󶀡GN (𝑥𝑥) , GN 󶀡󶀡𝑦𝑦󶀱󶀱 ,ℎ󶀱󶀱
GCGN(𝑒𝑒𝑒 󶀡󶀡GN (𝑥𝑥) , GN 󶀡󶀡𝑦𝑦󶀱󶀱 ,ℎ󶀱󶀱 = GN (𝑒𝑒) .
en,
= min 󶀧󶀧GN (𝑒𝑒) , (𝑏𝑏 𝑏 𝑏𝑏)
× 󶁦󶁦max 󶀦󶀦0,
−
(GN (𝑥𝑥) −𝑎𝑎)𝑚𝑚 + 󶀡󶀡GN 󶀡󶀡𝑦𝑦󶀱󶀱 −𝑎𝑎󶀱󶀱
(𝑏𝑏 𝑏 𝑏𝑏)𝑚𝑚
(GN (𝑒𝑒) −𝑎𝑎)𝑚𝑚
󶀶󶀶󶀶󶀶
(𝑏𝑏 𝑏 𝑏𝑏)𝑚𝑚
1/𝑚𝑚
𝑚𝑚
+ 𝑎𝑎󶀷󶀷 .
GC
(22)
Proof. Setting the interval [𝑎𝑎𝑎𝑎𝑎𝑎 of 𝑥𝑥𝑥 𝑥𝑥 as [0, 1], the lemma
can be proved simply.
�
𝑚𝑚
𝑚𝑚
󶀡󶀡𝑏𝑏 𝑏 𝑏𝑏󶀱󶀱 − (𝑏𝑏 𝑏 𝑏𝑏)𝑚𝑚 󶀅󶀅
+
󶀶󶀶
(𝑏𝑏 𝑏 𝑏𝑏)𝑚𝑚
󶀝󶀝
1/𝑚𝑚
+ 𝑎𝑎󶀺󶀺 .
(23)
GN 󶀢󶀢GCGN(𝑒𝑒𝑒 󶀡󶀡GN (𝑥𝑥) , GN 󶀡󶀡𝑦𝑦󶀱󶀱 ,ℎ󶀱󶀱󶀱󶀱
= GC𝑒𝑒 󶀡󶀡𝑥𝑥𝑥 𝑥𝑥𝑥 𝑥󶀱󶀱 .
(𝑏𝑏 𝑏 𝑏𝑏)𝑚𝑚
(𝑏𝑏 𝑏 𝑏𝑏)𝑚𝑚
𝑚𝑚
󶀡󶀡𝑏𝑏 𝑏 𝑏𝑏󶀱󶀱 − (𝑏𝑏 𝑏 𝑏𝑏)𝑚𝑚
+
󶀶󶀶󶁶󶁶
(𝑏𝑏 𝑏 𝑏𝑏)𝑚𝑚
= 2 (𝑏𝑏 𝑏 𝑏𝑏 𝑏 𝑏𝑏)
= 2GN (𝑒𝑒) .
GN (𝑥𝑥) = 𝑥𝑥∗ .
(28)
GN (𝑒𝑒) = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
(29)
From (28) and (29), the following could be obtained:
1/𝑚𝑚
+ 𝑎𝑎󶀷󶀷
(24)
GCGN(𝑒𝑒𝑒 󶀡󶀡GN (𝑥𝑥) , GN 󶀡󶀡𝑦𝑦󶀱󶀱 ,ℎ󶀱󶀱 = GC𝑒𝑒 󶀡󶀡𝑥𝑥∗ ,𝑦𝑦∗ ,ℎ󶀱󶀱 ,
(25)
∗
GN 󶀡󶀡GC𝑒𝑒 󶀡󶀡𝑥𝑥𝑥 𝑥𝑥𝑥 𝑥󶀱󶀱󶀱󶀱 = 󶀡󶀡GC𝑒𝑒 󶀡󶀡𝑥𝑥𝑥 𝑥𝑥𝑥 𝑥󶀱󶀱󶀱󶀱 .
And then from Lemma 6,
∗
GC𝑒𝑒 󶀡󶀡𝑥𝑥∗ ,𝑦𝑦∗ ,ℎ󶀱󶀱 = 󶀡󶀡GC𝑒𝑒 󶀡󶀡𝑥𝑥𝑥 𝑥𝑥𝑥 𝑥󶀱󶀱󶀱󶀱 .
So, the theorem is true.
GN (𝑥𝑥) + GN 󶀡󶀡𝑦𝑦󶀱󶀱
= 2 (𝑏𝑏 𝑏 𝑏𝑏) − 󶀡󶀡𝑥𝑥 𝑥 𝑥𝑥󶀱󶀱
Proof. Since interval [𝑎𝑎𝑎𝑎𝑎𝑎 relates to 𝑒𝑒 symmetry, then 𝑎𝑎𝑎𝑎𝑎𝑎
2𝑒𝑒, and GN(𝑥𝑥𝑥 𝑥𝑥𝑥𝑥𝑥𝑥𝑥 𝑥𝑥 𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥. us,
GN 󶀡󶀡𝑦𝑦󶀱󶀱 = 𝑦𝑦∗ ,
(3) When 𝑥𝑥 𝑥 𝑥𝑥 𝑥 𝑥𝑥𝑥,
= (𝑏𝑏 𝑏 𝑏𝑏 𝑏 𝑏𝑏) + 󶀡󶀡𝑏𝑏 𝑏 𝑏𝑏 𝑏 𝑏𝑏󶀱󶀱
Lemma 8. When interval [𝑎𝑎𝑎𝑎𝑎𝑎 relates to 𝑒𝑒 symmetry,
∗
𝐺𝐺𝐺𝐺𝑒𝑒 (𝑥𝑥∗ ,𝑦𝑦∗ ,ℎ) = (𝐺𝐺𝐺𝐺𝑒𝑒 (𝑥𝑥𝑥 𝑥𝑥𝑥 𝑥𝑥𝑥 , wherein 𝑥𝑥∗ represents the
points of 𝑥𝑥 relating to 𝑒𝑒 symmetry, namely, 𝑥𝑥∗ = 2𝑒𝑒 𝑒𝑒𝑒𝑒𝑒𝑒∗ ,
∗
𝑒𝑒∗ , (𝐺𝐺𝐺𝐺𝑒𝑒 (𝑥𝑥𝑥 𝑥𝑥𝑥 𝑥𝑥𝑥 is similar, 𝑒𝑒 𝑒 𝑒𝑒𝑒𝑒𝑒𝑒𝑒, ℎ∈ [0, 1].
Similarly,
= 𝑏𝑏 𝑏 𝑏𝑏 𝑏 𝑏𝑏𝑏 󶀧󶀧𝑏𝑏 𝑏 𝑏𝑏 𝑏 𝑏𝑏𝑏 (𝑏𝑏 𝑏 𝑏𝑏)
× 󶁥󶁥max 󶀥󶀥0,
Lemma 6. Let 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 (𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝑒𝑒 (𝑥𝑥𝑥 𝑥𝑥𝑥
ℎ)).
Lemma 7. Let 𝐶𝐶𝑒𝑒 (𝑥𝑥𝑥 𝑥𝑥𝑥 𝑥𝑥 𝑥𝑥𝑥𝑥𝑥1−𝑒𝑒 (1 − 𝑥𝑥𝑥 𝑥𝑥 𝑥𝑥𝑥 𝑥𝑥.
(𝑏𝑏 𝑏 𝑏𝑏)
× 󶀄󶀄max 󶀧󶀧0,
(𝑏𝑏 𝑏 𝑏𝑏)𝑚𝑚
󶀜󶀜
And then,
From the aforementioned, the theorem is true.
Proof. According to eorem 5 and involution law of central
generalized negation operator in interval [𝑎𝑎𝑎𝑎𝑎𝑎, the theorem
can be proved simply.
�
󶀡󶀡GN (𝑥𝑥) , GN 󶀡󶀡𝑦𝑦󶀱󶀱 ,ℎ󶀱󶀱
= min 󶀧󶀧𝑏𝑏 𝑏 𝑏𝑏 𝑏 𝑏𝑏𝑏 (𝑏𝑏 𝑏 𝑏𝑏)
GN 󶀢󶀢GCGN(𝑒𝑒𝑒 󶀡󶀡GN (𝑥𝑥) , GN 󶀡󶀡𝑦𝑦󶀱󶀱 ,ℎ󶀱󶀱󶀱󶀱 = GN (GN (𝑒𝑒)) = 𝑒𝑒𝑒
(27)
�
Substituting (16), (17), and (18) separately into (22),
GN(𝑒𝑒𝑒
(26)
(30)
(31)
�
Lemma 9. Let 𝐶𝐶0.5 (1 − 𝑥𝑥𝑥 𝑥𝑥 𝑥𝑥𝑥 𝑥𝑥 𝑥𝑥𝑥𝑥𝑥0.5 (𝑥𝑥𝑥 𝑥𝑥𝑥 𝑥𝑥, wherein
ℎ∈ [0, 1].
Lemma 10. When interval [𝑎𝑎𝑎𝑎𝑎𝑎 relates to symmetry of original point, 𝐶𝐶0 (−𝑥𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥 𝑥𝑥𝑥𝑥𝑥𝑥0 (𝑥𝑥𝑥 𝑥𝑥𝑥 𝑥𝑥, ℎ∈ [0, 1].
is lemma indicates that when the interval [𝑎𝑎𝑎𝑎𝑎𝑎 is
symmetrical about the origin point and identity element 𝑒𝑒
6
Discrete Dynamics in Nature and Society
2.3. Universal Combinatorial Operation Model with Unequal
Weights. In practical complex system, every factor is generally with unequal weight. But the existing universal combinatorial operation only discusses an ideal state that every factor
is with equal weight.
2.3.1. Weighted Operator. For dealing reasonably with the
complex relations of every factor in complex system, various
properties which weighted operators should have are put
forward. e weighted operator proposed by Yager is one of
the famous ones.
e weighted operator proposed by �ager is de�ned as
follows.
�e�nition 11 (see [12]). Assume that an operator ℎ(𝛼𝛼𝛼𝛼𝛼𝛼 is a
mapping from [0, 1] to [0, 1].
ℎ(𝛼𝛼𝛼𝛼𝛼𝛼 is called a Yager weighted operator if it satis�es the
following properties.
(I1) Monotonicity with respect to the value, 𝑥𝑥. In particular if 𝑥𝑥 𝑥 𝑥𝑥′ , then we require
ℎ (𝛼𝛼𝛼𝛼𝛼) ≥ ℎ 󶀣󶀣𝛼𝛼𝛼𝛼𝛼′ 󶀳󶀳 .
(32)
(I2) We desire that elements with weight zero have no
effect on the aggregation process; thus, if 𝑒𝑒 is the �xed
identity of the aggregation to be used on the resulting
bag, we must have
ℎ (0, 𝑥𝑥) = 𝑒𝑒𝑒
(33)
ℎ (1, 𝑥𝑥) = 𝑥𝑥𝑥
(34)
(I3) A normalcy with respect to the weights
(I4) Finally, we desire that the transformation moves
monotonically from its value for 𝛼𝛼 𝛼𝛼 to 𝛼𝛼 𝛼𝛼.
at is, if 𝑥𝑥 𝑥𝑥𝑥, ℎ(𝛼𝛼𝛼𝛼𝛼𝛼 increases monotonically with
respect to the value 𝛼𝛼; if 𝑥𝑥 𝑥 𝑥𝑥, ℎ(𝛼𝛼𝛼𝛼𝛼𝛼 decreases
monotonically with respect to the value 𝛼𝛼,
where 𝛼𝛼 is the weight associated with an argument, and 𝑥𝑥 is
the argument. Both 𝛼𝛼 and 𝑥𝑥 are drawn from [0, 1].
And due to the de�nition of Yager weighted operator, we
can draw the following theorems.
ℎ(𝛼, 𝑥1 )
𝑥1
𝛼=1
0
ℎ(𝛼, 𝑥2 )
𝑥2 1
𝛼=1
𝑒
𝛼=0
F 1: Yager weighted operator ℎ(𝛼𝛼𝛼𝛼𝛼𝛼 varies with the weight 𝛼𝛼.
1
0.9
0.8
0.7
0.6
ℎ(𝛼, 𝑥)
is 0, Universal Combinatorial Operation GC𝑒𝑒 (𝑥𝑥𝑥 𝑥𝑥𝑥 𝑥𝑥 is also
symmetrical about the origin point.
As pointed out in the literature [3], the internal relation
between fuzzy input variables, the deviation 𝐸𝐸, and the deviation change 𝐸𝐸𝐸𝐸 of normal two-dimensional fuzzy controller
is the universal combination relation in universal logic. Consequently, the complex fuzzy rule inference process could
be replaced by the simple universal combinatorial operation,
and a �exible logic control model was presented accordingly.
In fuzzy control, the domain of input variables and output
variable is generally symmetrical about the original point,
such as [−5, 5]. Obviously, it is the prerequisite of control
model that the operation model relates to symmetry of
original point. erefore, Lemma 10 provides a basis for the
Universal Combinatorial Operation’s application in control.
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
𝑥
𝛼=0
𝛼 = 0.1
𝛼 = 0.3
𝛼 = 0.5
𝛼 = 0.7
𝛼 = 0.9
𝛼=1
F 2: Yager weighted operator ℎ(𝛼𝛼𝛼𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼 𝛼 𝛼𝛼𝛼 𝛼𝛼𝛼𝛼𝛼 with 𝑒𝑒 is
equal to 0.3.
eorem 12. Assume that ℎ(𝛼𝛼𝛼𝛼𝛼𝛼 is a Yager weighted operator. So, one has the following:
if 𝑥𝑥 𝑥𝑥𝑥𝑥 then ℎ (𝛼𝛼𝛼𝛼𝛼) ≥ 𝑒𝑒𝑒 and if 𝑥𝑥 𝑥 𝑥𝑥𝑥 then ℎ (𝛼𝛼𝛼𝛼𝛼) ≤ 𝑒𝑒𝑒
(35)
Proof. Due to the properties (I2) and (I4) of the de�nition,
the theorem can be proved easily.
�
eorem 13. Assume that ℎ(𝛼𝛼𝛼𝛼𝛼𝛼 is a Yager weighted operator. So, one has the following:
if 𝑥𝑥 𝑥𝑥𝑥𝑥 then ℎ (𝛼𝛼𝛼𝛼𝛼) ≥ 𝑒𝑒𝑒 and if 𝑥𝑥 𝑥 𝑥𝑥𝑥 then ℎ (𝛼𝛼𝛼𝛼𝛼) ≤ 𝑒𝑒𝑒
(36)
Proof. Due to the properties (I3) and (I4) of the de�nition,
the theorem can be proved easily.
e chart of Yager weighted operator ℎ(𝛼𝛼𝛼𝛼𝛼𝛼 changing
along with the weight 𝛼𝛼 is given in Figure 1.
One formulation that satis�es these conditions is
ℎ(𝛼𝛼𝛼𝛼𝛼𝛼𝛼 𝛼𝛼𝛼𝛼 𝛼 𝛼𝛼𝛼 𝛼𝛼𝛼𝛼𝛼. It is described by Figure 2.
According to the previous analyses, we discover that Yager
weighted operator has some shortcomings as follows.
Discrete Dynamics in Nature and Society
Gh(𝑎, 𝑥1 )
7
Gh(𝑎, 𝑥1 )
1
0.8
𝛼=1
𝑒
𝛼=0
𝑥2
𝑏
0.6
𝛼=1
F 3: General weighted operator varies with the weight 𝛼𝛼.
(1) Yager weighted operator is likely to transform entire
True (False) proposition into partial True (False)
proposition. However, due to the view of logic, entire
True (False) proposition should still be transformed
into entire True (False) proposition by weighted
operator.
(2) e weighted value changes in the interval [𝑒𝑒𝑒 𝑒𝑒𝑒 or
[𝑥𝑥𝑥𝑥𝑥𝑥 for any weight 𝛼𝛼 𝛼 𝛼𝛼𝛼 𝛼𝛼 as shown in Figure 1.
However, the weighted value is desired to change in
the total interval [0,1] in some practical applications.
(3) Yager weighted operator is limited in the interval [0,1].
But weighted operators are desired to change in the
general interval [𝑎𝑎𝑎 𝑎𝑎𝑎 in some practical applications.
For solving the previous problems, the paper puts forward
a kind of general weighted operators Gh(𝛼𝛼𝛼𝛼𝛼𝛼, which change
in the general interval [𝑎𝑎𝑎 𝑎𝑎𝑎.
�
�e�nition ��. Assume that an operator Gh(𝛼𝛼𝛼𝛼𝛼𝛼 is a mapping
from [𝑎𝑎𝑎 𝑎𝑎𝑎 to [𝑎𝑎𝑎 𝑎𝑎𝑎. Gh(𝛼𝛼𝛼𝛼𝛼𝛼 is called a general weighted
operator if it satis�es the following properties.
(I1) Monotonicity with respect to the value, 𝑥𝑥. In particular, if 𝑥𝑥 𝑥 𝑥𝑥′ , then we require
Gh (𝛼𝛼𝛼𝛼𝛼) ≥ Gh 󶀣󶀣𝛼𝛼𝛼𝛼𝛼′ 󶀳󶀳 .
(37)
(I2) Gh(0,𝑥𝑥𝑥 𝑥 ite{𝑎𝑎 𝑎 𝑎𝑎𝑎 𝑎𝑎𝑎 𝑎𝑎 𝑎 𝑎𝑎𝑎 𝑎𝑎𝑎 𝑎𝑎𝑎.
(I3) Gh(𝛼𝛼𝛼𝛼𝛼𝛼 𝛼𝛼𝛼.
(I4) Gh(𝛼𝛼𝛼𝛼𝛼𝛼 𝛼𝛼𝛼 and Gh(𝛼𝛼𝛼𝛼𝛼𝛼 𝛼𝛼𝛼.
(I5) Finally, we desire that the transformation moves
monotonically for the weight 𝛼𝛼 𝛼 𝛼𝛼𝛼 𝛼𝛼. at is,
if 𝑥𝑥𝑥𝑥𝑥, Gh(𝛼𝛼𝛼𝛼𝛼𝛼 increases monotonically with
respect to the value 𝛼𝛼; if 𝑥𝑥 𝑥 𝑥𝑥, Gh(𝛼𝛼𝛼𝛼𝛼𝛼 decreases
monotonically with respect to the value 𝛼𝛼.
(I6) Gh(1,𝑥𝑥𝑥 𝑥 ite{𝑒𝑒 𝑒 𝑒𝑒 𝑒 𝑒𝑒𝑒𝑒𝑒𝑒 𝑒𝑒 𝑒 𝑒𝑒𝑒𝑒𝑒𝑒 𝑒𝑒 𝑒 𝑒𝑒𝑒.
𝛼𝛼 is the weight associated with an argument 𝑥𝑥, and 𝑒𝑒 is the
��ed identity of the aggregation operator GC𝑒𝑒 . 𝛼𝛼 is drawn
from [0,1], and 𝑒𝑒 is drawn from [𝑎𝑎𝑎 𝑎𝑎𝑎.
According to the previous de�nition, we can know that
the absolute value of the argument 𝑥𝑥 decreases �rst and then
increases with the weight 𝛼𝛼 changing continuously from 0 to
1. e chart of general weighted operator changing along with
the weight 𝛼𝛼 is given in Figure 3.
0.4
0.2
𝐺ℎ(𝛼, 𝑥)
𝑥1
𝑎
0
−0.2
−0.4
−0.6
−0.8
−1
−1
−0.8 −0.6 −0.4 −0.2
0
0.2
0.4
0.6
0.8
1
𝑥
𝛼=0
𝛼 = 0.1
𝛼 = 0.3
𝛼 = 0.5
𝛼 = 0.7
𝛼 = 0.9
𝛼=1
F 4: General weighted operator of polynomial model with 𝑎𝑎 𝑎
−1, 𝑏𝑏𝑏𝑏, and 𝑒𝑒 𝑒𝑒𝑒𝑒𝑒.
e common general weighted operation model is polynomial model:
Gh (𝛼𝛼𝛼𝛼𝛼)
= ite 󶁇󶁇
(𝑥𝑥𝑥𝑥𝑥) (𝑏𝑏𝑏𝑏𝑏) (1 + 𝑛𝑛)1/2
+ 𝑒𝑒 𝑒 𝑒𝑒 𝑒 𝑒𝑒𝑒
󶀢󶀢(𝑏𝑏𝑏𝑏𝑏) + (1 + 𝑛𝑛)1/2 (𝑥𝑥𝑥𝑥𝑥)󶀲󶀲
𝑒𝑒 𝑒
(𝑒𝑒 𝑒 𝑒𝑒) (𝑒𝑒 𝑒𝑒𝑒) (1 + 𝑛𝑛)1/2
∣𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥󶁗󶁗 .
󶀢󶀢(𝑥𝑥𝑥𝑥𝑥) + (1 + 𝑛𝑛)1/2 (𝑒𝑒 𝑒 𝑒𝑒)󶀲󶀲
(38)
Assume that 𝑛𝑛 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛2 , 𝛼𝛼 𝛼 𝛼𝛼𝛼 𝛼𝛼, 𝑛𝑛 𝑛𝑛𝑛, and
Gh(𝛼𝛼𝛼𝛼𝛼𝛼 is the limit as 𝛼𝛼 𝛼 𝛼 and 𝛼𝛼 𝛼 𝛼. So, (38) describes a
general weighted operator as shown in Figure 4.
2.3.2. Universal Combinatorial Operation Model with Unequal Weights. According to the previous de�nition of general weighted operators, we can get the de�nition of universal
combinatorial operation model with unequal weights as follows.
�e�nition ��. Assume that an operator UGC𝑒𝑒 (𝑥𝑥𝑥 𝑥𝑥𝑥𝑥𝑥𝑥𝑥 , 𝛼𝛼𝑦𝑦 , ℎ)
is a mapping from [𝑎𝑎𝑎 𝑎𝑎𝑎 to [𝑎𝑎𝑎 𝑎𝑎𝑎. UGC𝑒𝑒 (𝑥𝑥𝑥 𝑥𝑥𝑥𝑥𝑥𝑥𝑥 , 𝛼𝛼𝑦𝑦 , ℎ) is
called universal combinatorial operation model with unequal
weights:
UGC𝑒𝑒 󶀢󶀢𝑥𝑥𝑥 𝑥𝑥𝑥𝑥𝑥𝑥𝑥 , 𝛼𝛼𝑦𝑦 , ℎ󶀲󶀲 = GC𝑒𝑒 󶀢󶀢Gh 󶀡󶀡𝛼𝛼𝑥𝑥 ,𝑥𝑥󶀱󶀱 , Gh 󶀢󶀢𝛼𝛼𝑦𝑦 ,𝑦𝑦󶀲󶀲 , ℎ󶀲󶀲 ,
(39)
8
Discrete Dynamics in Nature and Society
T 2: Fuzzy rules de�ning fuzzy composed variable 𝐺𝐺𝐺𝐺𝜃𝜃 .
𝑈𝐺𝐶𝑒(𝑥, 𝑦, 𝛼𝑥 , 𝛼𝑦 , ℎ)
𝑥
𝐺ℎ(𝛼𝑥 , 𝑥)
𝑥󳰀
𝐺𝐶𝑒 (𝑥󳰀 , 𝑦󳰀 , ℎ)
𝑦
𝐺ℎ(𝛼𝑦 , 𝑦)
𝐺𝐸𝜃
𝑧
𝑦󳰀
F 5: Universal combinatorial operation model with unequal
weights.
where Gh(𝛼𝛼𝛼 𝛼𝛼𝛼 is general weighted operator, GC𝑒𝑒 (𝑥𝑥𝑥 𝑥𝑥𝑥 𝑥𝑥 is
universal combinatorial operator, 𝑒𝑒 is the �xed identity of
GC𝑒𝑒 (𝑥𝑥𝑥 𝑥𝑥𝑥 𝑥𝑥, ℎ is the generalized correlation coefficient, and
𝛼𝛼𝑥𝑥 and 𝛼𝛼𝑦𝑦 denote, respectively, the weights associated with
the arguments 𝑥𝑥 and 𝑦𝑦. 𝑥𝑥, 𝑦𝑦, and 𝑒𝑒 are drawn from [𝑎𝑎𝑎 𝑎𝑎𝑎, and
ℎ, 𝛼𝛼𝑥𝑥 , and 𝛼𝛼𝑦𝑦 are drawn from [0, 1].
e universal combinatorial operation model with unequal weights is given in Figure 5.
2.4. Flexible Logic Control Method. In order to decrease effectively the number of fuzzy control rules for multivariable
nonlinear system, Xiao et al. [13] gave a new concept of
fuzzy composed variable. Its basic idea can be summarized as
follows. According to characteristics of controlled system and
internal relations between input variables, a fuzzy composed
variable is constructed to re�ect synthetically the deviation
between reference and the process output with a fuzzy logic
system.
ere are four output variables in single inverted-pendulum, which are 𝑥𝑥𝑥𝑥𝑥′ , 𝜃𝜃, and 𝜃𝜃′ . In the four input variables of
control system, it is 𝜃𝜃 and 𝜃𝜃′ that describe the movement state
of the rod. So, a fuzzy logic system can be designed, which
is described by the fuzzy rules in Table 2, to de�ne a fuzzy
composed variable 𝐺𝐺𝐺𝐺𝜃𝜃 with 𝜃𝜃 and 𝜃𝜃′ . e fuzzy composed
variable 𝐺𝐺𝐺𝐺𝜃𝜃 can describe synthetically the movement state
of the rod. Similarly, a fuzzy composed variable 𝐺𝐺𝐺𝐺𝑥𝑥 can be
de�ned with 𝑥𝑥 and 𝑥𝑥′ to describe synthetically the movement
state of the cart. For multivariable system, one does not need
to de�ne, respectively, fuzzy logic system for every fuzzy
composed variable. A uniform fuzzy rule table can be used,
such as Table 2, but only select different quanti�cation factors
to obtain different fuzzy composed variables.
Remark 16. In the paper, the input variables of the fuzzy
controllers discussed are 𝑒𝑒 and 𝑒𝑒𝑒𝑒, which denote, respectively,
the deviation and the deviation change. e output variable 𝑢𝑢
is the control signal. e variables, 𝑒𝑒, 𝑒𝑒𝑒𝑒, and 𝑢𝑢 are crisp values
from the practical process. e fuzzy variables, 𝐸𝐸, 𝐸𝐸𝐸𝐸, and 𝑈𝑈
are the corresponding fuzzy ones, and the fuzzy domains are
uniformed to be [−1, 1] with fuzzy subsets, such as NB, NM,
NS, ZE, PS, PM, and PB.
Apparently, the fuzzy rules in Table 2 describe the
essential relation between all deviation and the deviation
NB
NM
NS
𝜃 ZE
PS
PM
PB
NB
NB
NB
NB
NM
NM
NS
ZE
NM
NB
NB
NM
NM
NS
ZE
PS
NS
NB
NM
NM
NS
ZE
PS
PM
𝜃󳰀
ZE
NM
NM
NS
ZE
PS
PM
PM
PS
NM
NS
ZE
PS
PM
PM
PB
PM
NS
ZE
PS
PM
PM
PB
PB
PB
ZE
PS
PM
PM
PB
PB
PB
change. As shown Table 2 can be divided approximately into
four parts, which describe, respectively, fuzzy rules used to
de�ne composed variable as 𝐸𝐸 and 𝐸𝐸𝐸𝐸 are both negative, 𝐸𝐸 is
negative but 𝐸𝐸𝐸𝐸 is positive, 𝐸𝐸 and 𝐸𝐸𝐸𝐸 are both positive, and
𝐸𝐸 is positive but 𝐸𝐸𝐸𝐸 is negative. 𝐸𝐸 and 𝐸𝐸𝐸𝐸 both describe the
deviation between the reference and the process output; so,
we can de�ne a composed variable, denoted as 𝐸𝐸′ , based on
the essential relation between them.
According to the physical meanings of fuzzy variables, 𝐸𝐸
and 𝐸𝐸𝐸𝐸, we can get the following conclusions.
(1) Suppose that both 𝐸𝐸 and 𝐸𝐸𝐸𝐸 are positive. at is
to say, the deviation is positive and it will increase
continuously. So, the value of the composed variable
𝐸𝐸′ should be positive and bigger than both of 𝐸𝐸 and
𝐸𝐸𝐸𝐸 in this case. e combinatorial rules are shown by
the right-bottom part of Table 2.
(2) Suppose that 𝐸𝐸 is positive and 𝐸𝐸𝐸𝐸 is negative. at
is to say, the deviation is positive but it will decrease.
So, the value of the composed variable 𝐸𝐸′ should be
between 𝐸𝐸 and 𝐸𝐸𝐸𝐸 in this case. e combinatorial
rules are shown by the le-bottom part of Table 2.
(3) Similarly, we can obtain the value of 𝐸𝐸′ in the two of
other cases.
Based on the previous analysis, we get the conclusion
that the essential relation among 𝐸𝐸, 𝐸𝐸𝐸𝐸, and the composed
variable 𝐸𝐸′ is a kind of universal combinatorial one in
Universal Logic. As a result, we have
𝐸𝐸′ = GC𝑒𝑒 (𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸) .
(40)
𝑈𝑈 𝑈𝑈GC𝑒𝑒 (𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸) ,
(41)
So, we can obtain the relation among 𝐸𝐸, 𝐸𝐸𝐸𝐸, and the
output variable 𝑈𝑈 of control system as follows:
where fuzzy variables, 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐸 𝐸𝐸𝐸𝐸𝐸𝐸, 𝑒𝑒 𝑒𝑒, ℎ∈ [0, 1].
Fu and He [3] named the method Flexible Logic Control
Method.
Discrete Dynamics in Nature and Society
9
F 6: e �exible logic control model of single inverted-pendulum.
For effectively controlling different things, we can lead
into a weighted factor 𝛼𝛼𝛼 𝛼𝛼 𝛼 𝛼𝛼𝛼 𝛼𝛼. As the correlation
coefficient ℎ is equal to 0.5, we have
𝑈𝑈 𝑈 𝑈GC𝑒𝑒 (𝛼𝛼𝛼𝛼𝛼 (1− 𝛼𝛼) 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸)
=
Γ1−1
[𝛼𝛼𝛼𝛼 𝛼 (1− 𝛼𝛼) 𝐸𝐸𝐸𝐸 𝐸𝐸𝐸]
(42)
= 𝛼𝛼𝛼𝛼 𝛼 (1− 𝛼𝛼) 𝐸𝐸𝐸𝐸𝐸
However, (42) is just the fuzzy control method proposed by
Long et al. [14]. ey used a linear equation, such as (42), to
describe fuzzy control rules. But the relation among 𝐸𝐸, 𝐸𝐸𝐸𝐸,
and 𝑈𝑈 is not only linear. So, (41) is a cluster of operators
determined by correlation coefficient ℎ, and (42) is only a
special operator in the cluster as ℎ is equal to 0.5. As a result,
�exible logic control method can realize the accurate control
for complex system.
2.5. Flexible Logic Control Model of Single Inverted-Pendulum
System. e objective is to maintain the pole in an upright
position and the cart in an appointed position in the rail.
ere are four output variables and one input variable in
single inverted-pendulum, which are 𝑥𝑥, 𝑥𝑥′ , 𝜃𝜃, 𝜃𝜃′ , and 𝑢𝑢. In the
four input variables of control system, both 𝜃𝜃 and 𝜃𝜃′ describe
the movement state of the pole, and both 𝑥𝑥 and 𝑥𝑥′ describe
the movement state of the cart. So, we have designed two
subcontrollers. One is to maintain the cart in an appointed
position with two input variables 𝐸𝐸𝑥𝑥 and 𝐸𝐸𝐸𝐸𝑥𝑥 . e other one
is to maintain the pole in an upright position with two input
variables 𝐸𝐸𝜃𝜃 and 𝐸𝐸𝐸𝐸𝜃𝜃 .
We have designed the two subcontrollers with the �exible
logic control method. And we led into weighted factors 𝛼𝛼𝜃𝜃 ,
𝛼𝛼𝑥𝑥 , 𝛼𝛼𝑥𝑥 , 𝛼𝛼𝜃𝜃 ∈[0,1]. Two controllers are designed as follows:
𝑈𝑈𝜃𝜃 =−GC0 󶀡󶀡Gh 󶀡󶀡𝛼𝛼𝜃𝜃 ,𝐸𝐸𝜃𝜃 󶀱󶀱 , Gh 󶀡󶀡1− 𝛼𝛼𝜃𝜃 ,𝐸𝐸𝐸𝐸𝜃𝜃 󶀱󶀱 , ℎ𝜃𝜃 󶀱󶀱 ,
(43)
𝑈𝑈𝑥𝑥 =−GC0 󶀡󶀡Gh 󶀡󶀡𝛼𝛼𝑥𝑥 ,𝐸𝐸𝑥𝑥 󶀱󶀱 , Gh 󶀡󶀡1− 𝛼𝛼𝑥𝑥 ,𝐸𝐸𝐸𝐸𝑥𝑥 󶀱󶀱 , ℎ𝑥𝑥 󶀱󶀱 ,
(45)
𝛼𝛼𝜃𝜃 = 󶀡󶀡𝛼𝛼𝑠𝑠_𝜃𝜃 − 𝛼𝛼0_𝜃𝜃 󶀱󶀱 󶀱󶀱𝐸𝐸𝜃𝜃 󶙡󶙡 + 𝛼𝛼0_𝜃𝜃 ,
(44)
𝛼𝛼𝑥𝑥 = 󶀡󶀡𝛼𝛼𝑠𝑠_𝑥𝑥 − 𝛼𝛼0_𝑥𝑥 󶀱󶀱 󶙡󶙡𝐸𝐸𝑥𝑥 󶙡󶙡 + 𝛼𝛼0_𝑥𝑥 .
(46)
When the control signal 𝑈𝑈 is combined, we lead into
two weighted factors 𝑘𝑘𝜃𝜃 , 𝑘𝑘𝑥𝑥 , 𝑘𝑘𝜃𝜃 , 𝑘𝑘𝑥𝑥 ∈[−1,1], for the two
subcontrollers. According to (43) and (45), we can get the
output of the controller as follows:
𝑈𝑈 𝑈 𝑈𝑈𝑥𝑥 𝑈𝑈𝑥𝑥 + 𝑘𝑘𝜃𝜃 𝑈𝑈𝜃𝜃
= 𝑘𝑘𝑥𝑥 󶀢󶀢−GC0 󶀡󶀡Gh 󶀡󶀡𝛼𝛼𝑥𝑥 ,𝐸𝐸𝑥𝑥 󶀱󶀱 , Gh 󶀡󶀡1− 𝛼𝛼𝑥𝑥 ,𝐸𝐸𝐸𝐸𝑥𝑥 󶀱󶀱 , ℎ𝑥𝑥 󶀱󶀱󶀱󶀱
+ 𝑘𝑘𝜃𝜃 󶀢󶀢−GC0 󶀡󶀡Gh 󶀡󶀡𝛼𝛼𝜃𝜃 ,𝐸𝐸𝜃𝜃 󶀱󶀱 , Gh 󶀡󶀡1− 𝛼𝛼𝜃𝜃 ,𝐸𝐸𝐸𝐸𝜃𝜃 󶀱󶀱 , ℎ𝜃𝜃 󶀱󶀱󶀱󶀱 ,
(47)
where 𝐸𝐸𝑥𝑥 ,𝐸𝐸𝐸𝐸𝑥𝑥 , 𝑈𝑈𝑥𝑥 ,𝐸𝐸𝜃𝜃 ,𝐸𝐸𝐸𝐸𝜃𝜃 , 𝑈𝑈𝜃𝜃 , 𝑈𝑈 𝑈𝑈𝑈𝑈𝑈𝑈𝑈, Gh(𝛼𝛼𝛼 𝛼𝛼𝛼 (38),
UGC0 (𝑥𝑥𝑥 𝑥𝑥𝑥𝑥𝑥1 , 𝛼𝛼2 , ℎ) is (39), ℎ𝑥𝑥 , ℎ𝜃𝜃 are general correlation
coefficients, ℎ𝑥𝑥 , ℎ𝜃𝜃 ∈[0,1], 𝛼𝛼𝑥𝑥 , 𝛼𝛼𝜃𝜃 ∈[0,1], 0 ≤ 𝛼𝛼0_𝑥𝑥 ≤ 𝛼𝛼𝑠𝑠_𝑥𝑥 ≤
1, and 0 ≤ 𝛼𝛼0_𝜃𝜃 ≤ 𝛼𝛼𝑠𝑠_𝜃𝜃 ≤ 1.
By the previous analyses, we can obtain the control model
of single inverted-pendulum as shown in Figure 6.
3. Parametric Analysis of Flexible
Logic Control Model
rough the previous analysis, it is clear that �exible ability
of �exible logic control model is resulted from the following
aspect. Universal combinatorial operation model is not a single �xed operator, but a continuous cluster of combinatorial
operators determined by the general correlation coefficient
ℎ between propositions. In practical control application,
according to the general correlation between propositions,
we can take the corresponding one from the cluster to realize
effective control for complex system.
However, in practical control application, how to determine the general correlation coefficient ℎ of �exible logic
control model is a problem for further studies. In this section,
we will analyze the general correlation coefficients, ℎ𝑥𝑥 , ℎ𝜃𝜃 ,
of the �exible logic control model and give some research
results.
3.1. Experimentation. We experiment the �exible logic control model in some single inverted-pendulum physical system. e physical parameters of the system are given in Table
3.
By simulating the inverted-pendulum and looking up the
optimization with genetic algorithm, we can get the initial
values of the control parameters. en, by testing in real-time
experimentations and repeatedly making some �ne tuning,
10
Discrete Dynamics in Nature and Society
T 3: Physical parameters of the quadruple inverted-pendulum.
Symbol
Value
Meaning
0.924 kg
See Table 5
𝑚𝑚0
𝑚𝑚1
Mass of the cart
Mass of the pole
Dynamic friction coefficient between the
0.1 N⋅s/m
cart and the track
0.007056 N⋅s/m Dynamic friction coefficient for the pole
See Table 5 Distance from the position sensor to the
center of gravity of the pole
𝑓𝑓0
𝑓𝑓1
𝑙𝑙1
T 6: e optimization results of ℎ𝑥𝑥 , ℎ𝜃𝜃 .
No.
1
2
3
4
5
6
7
8
𝑚𝑚1 (kg)
0.0149
0.0216
0.0284
0.0378
0.0493
0.0621
0.0773
0.0966
𝐿𝐿1 (m)
0.1
0.149
0.199
0.266
0.354
0.4436
0.553
0.691
ℎ𝑥𝑥
0.985
0.975
0.96
0.955
0.63
0.37
0.32
0.37
Fitness
0.4310
0.3310
0.2713
0.2240
0.1838
0.1531
0.1260
0.1029
ℎ𝜃𝜃
0.845
0.845
0.845
0.87
0.88
0.875
0.865
0.85
T 4: Control parameters of the control model.
Symbol Value Meaning
𝐾𝐾𝑒𝑒_𝑥𝑥
𝐾𝐾𝑐𝑐_𝑥𝑥
𝛼𝛼0_𝑥𝑥
𝛼𝛼𝑠𝑠_𝑥𝑥
𝐾𝐾𝑒𝑒_𝜃𝜃
𝐾𝐾𝑐𝑐_𝜃𝜃
𝛼𝛼0_𝜃𝜃
𝛼𝛼𝑠𝑠_𝜃𝜃
𝑘𝑘𝑥𝑥
𝑘𝑘𝜃𝜃
𝐾𝐾𝑢𝑢
ℎ𝑥𝑥
ℎ𝜃𝜃
23.5294
9.4118
0.1725
0.4683
50.5882
3.1765
0.2510
0.5953
−0.2235
0.4902
8.7843
�uanti�cation factor for 𝐸𝐸𝑥𝑥
�uanti�cation factor for 𝐸𝐸𝐸𝐸𝑥𝑥
Minimum value of 𝛼𝛼𝑥𝑥
Maximum value of 𝛼𝛼𝑥𝑥
�uanti�cation factor for 𝐸𝐸𝜃𝜃
�uanti�cation factor for 𝐸𝐸𝐸𝐸𝜃𝜃
Minimum value of 𝛼𝛼𝜃𝜃
Maximum value of 𝛼𝛼𝜃𝜃
Weighted factor of the subcontroller for the cart
Weighted factor of the subcontroller for the pole
Proportion factor for 𝑈𝑈
General correlation coefficient between 𝐸𝐸𝑥𝑥 and
0.2118
𝐸𝐸𝐸𝐸𝑥𝑥
General correlation coefficient between 𝐸𝐸𝜃𝜃 and
0.9490
𝐸𝐸𝐸𝐸𝜃𝜃
T 5: e variation of the length of the pole.
No.
1
2
3
4
5
6
7
8
𝑚𝑚1 (kg)
0.0149
0.0216
0.0284
0.0378
0.0493
0.0621
0.0773
0.0966
𝐿𝐿1 (m)
0.1
0.149
0.199
0.266
0.354
0.4436
0.553
0.691
we can get the control parameters shown in Table 4. And the
variation of the length of the pole is shown in Table 5.
Genetic algorithm is used to optimize the general correlation coefficients, ℎ𝑥𝑥 , ℎ𝜃𝜃 , and the �tness function is de�ned
as follows:
𝑁𝑁
Dis = 󵠈󵠈
𝑖𝑖𝑖𝑖
𝑖𝑖2 × 󶀣󶀣𝑥𝑥2 (𝑖𝑖) /10 + 𝑥𝑥′2 (𝑖𝑖) /20 + 𝜃𝜃2 (𝑖𝑖) + 𝜃𝜃′2 (𝑖𝑖)󶀳󶀳
�tness =
𝑁𝑁
1
.
(Dis/10)
,
(48)
rough simulated optimization, we can get the corresponding values of ℎ𝑥𝑥 , ℎ𝜃𝜃 and the �tness shown in Table 6.
Figure 7 shows how the �tness varies with ℎ𝑥𝑥 , ℎ𝜃𝜃 . And
Figures 8 and 9 illustrate how the maximum �tness varies
with ℎ𝜃𝜃 taking different values when ℎ𝑥𝑥 , in turn, is equal to
some value on standard interval [0, 1]. Figure 8 shows how
the maximum �tness varies with ℎ𝑥𝑥 , and Figure 9 shows how
ℎ𝜃𝜃 varies with ℎ𝑥𝑥 when the �tness is the maximum.
Similarly, Figures 10 and 11 illustrate how the maximum
�tness varies with ℎ𝑥𝑥 taking different values when ℎ𝜃𝜃 , in turn,
is equal to some value on standard interval [0, 1]. Figure 10
shows how the maximum �tness varies with ℎ𝜃𝜃 , and Figure 11
shows how ℎ𝑥𝑥 varies with ℎ𝜃𝜃 when the �tness is the maximum.
3.2. Parametric Analysis of Flexible Logic Control Model.
Firstly, calculate the correlation coefficient between �tness
and ℎ𝑥𝑥 when ℎ𝜃𝜃 , in turn, is equal to some value on standard
interval [0, 1]. e formula for calculating the correlation
coefficient is de�ned as follows:
𝑟𝑟 𝑟
∑𝑚𝑚 ∑𝑛𝑛 󶀢󶀢𝐴𝐴𝑚𝑚𝑚𝑚 − 𝐴𝐴󶀲󶀲 󶀲󶀲𝐵𝐵𝑚𝑚𝑚𝑚 − 𝐵𝐵󶀲󶀲
2
2
󵀊󵀊󶀣󶀣∑𝑚𝑚 ∑𝑛𝑛 󶀢󶀢𝐴𝐴𝑚𝑚𝑚𝑚 − 𝐴𝐴󶀲󶀲 󶀳󶀳 󶀳󶀳∑𝑚𝑚 ∑𝑛𝑛 󶀢󶀢𝐵𝐵𝑚𝑚𝑚𝑚 − 𝐵𝐵󶀲󶀲 󶀳󶀳
,
(49)
where 𝐴𝐴 and 𝐵𝐵 are 𝑚𝑚 𝑚 𝑚𝑚 matrices, 𝐴𝐴 and 𝐵𝐵 are the means
of the values in 𝐴𝐴 and 𝐵𝐵, respectively, and 𝑟𝑟 is the correlation
coefficient between 𝐴𝐴 and 𝐵𝐵.
Figure 12 illustrates how the correlation coefficient
between �tness and ℎ𝑥𝑥 varies with ℎ𝜃𝜃 taking different values
on standard interval [0, 1].
Similarly, calculate the correlation coefficient between
�tness and ℎ𝜃𝜃 when ℎ𝑥𝑥 , in turn, is equal to some value on
standard interval [0, 1]. And Figure 13 illustrates how the
correlation coefficient between �tness and ℎ𝜃𝜃 varies with ℎ𝑥𝑥
taking different values on standard interval [0, 1].
Aer that, the correlation coefficients between the maximum �tness and ℎ𝑥𝑥 , ℎ𝜃𝜃 are computed for poles of different
lengths, as shown in Table 7.
By observing and analyzing the previous results, we can
draw the following conclusions.
(1) e longer the pole, the worse the control effects.
Figures 7, 8, and 10 all indicate that the longer the
pole, the smaller the �tness with the optimum control
parameters, and vice versa. rough experiment, we
can know that the given single inverted-pendulum
Discrete Dynamics in Nature and Society
11
Fitness
0.45
0
1
1
0
ℎ𝜃
1
1
0
ℎ𝜃
ℎ𝑥
Fitness
(a)
1
1
1
ℎ𝑥
0
ℎ𝜃
(b)
1
ℎ𝑥
0
ℎ𝜃
(c)
ℎ𝑥
(d)
0.45
0
1
1
0
ℎ𝜃
1
1
ℎ𝑥
0
ℎ𝜃
(e)
1
1
ℎ𝑥
0
ℎ𝜃
(f)
ℎ𝑥
1
1
0
ℎ𝜃
(g)
ℎ𝑥
(h)
F 7: e �tness varies with ℎ𝑥𝑥 , ℎ𝜃𝜃 .
0.45
0.4
0.35
ℎ𝜃
Fitness
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.1
1
2
3
4
0.2
0.3
0.4
0.5
ℎ𝑥
0.6
0.7
0.8
0.9
1
5
6
7
8
F 8: e ma�imum �tness varies with ℎ𝑥𝑥 .
physical system can be controlled effectively when
the pole is in the range 0.1 m to 0.691 m. at is to
say, when the pole is longer than 0.691 m or shorter
than 0.1 m, we cannot realize the stable control for the
given physical system.
(2) For some given physical system, the control effect is
sensitive to the value of ℎ𝜃𝜃 . is means that we can
realize the effective control only when ℎ𝜃𝜃 is in some
very short interval. And the interval is relatively ��ed.
at is, the interval of ℎ𝜃𝜃 does not change with the
length of the pole. Figures 7, 9 and 10 all show that the
�tness is relatively big only when ℎ𝜃𝜃 is in the range 0.8
to 0.9, and the interval of ℎ𝜃𝜃 remains unchanged for
poles of different lengths.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
ℎ𝑥
0.6
0.7
0.8
0.9
1
F 9: ℎ𝜃𝜃 varies with ℎ𝑥𝑥 when the �tness is the ma�imum.
Figure 12 also shows that the correlation coefficient
between �tness and ℎ𝑥𝑥 is relatively big only when ℎ𝜃𝜃
is around 0.2 or in the range 0.8 to 0.9, and the interval
of ℎ𝜃𝜃 remains unchanged for poles of different lengths.
(3) For some given physical system, the control effect is
not sensitive to the value of ℎ𝑥𝑥 . at is, we can realize
the effective control when ℎ𝑥𝑥 is in some very long
interval. Figure 10 shows that the control effect does
not change much with ℎ𝑥𝑥 taking different values of the
long interval.
Figure 13 also shows that the correlation coefficient
between �tness and ℎ𝜃𝜃 is relatively big when ℎ𝑥𝑥 is in
the range 0.15 to 0.98.
However, the width of the interval varies with the
length of pole. e longer the pole, the narrower
the interval. e paper calls the interval of ℎ𝑥𝑥 as
ℎ𝑥𝑥 platform.
(4) For poles of different lengths, there is much difference
of ℎ𝑥𝑥 and little one of ℎ𝜃𝜃 when we realize the most
effective control. As shown in Table 7, the correlation
12
Discrete Dynamics in Nature and Society
0.85
0.45
Width of ℎ𝑥 platform
0.4
0.35
Fitness
0.3
0.25
0.2
0.15
0.1
0.8
0.75
0.7
0.05
0
0
0.1
0.2
0.3
0.4
0.5
ℎ𝜃
0.6
0.8
0.9
0.1
0.2
0.3
0.4
0.5
ℎ𝜃
0.6
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.7
0.8
0.9
1
F 12: e correlation coefficient between �tness and ℎ𝑥𝑥 varies
with ℎ𝜃𝜃 .
0
0.1
0.2
0.3
0.4
0.5
ℎ𝑥
0.6
6
7
T 7: e correlation coefficients between the maximum �tness
and ℎ𝑥𝑥 , ℎ𝜃𝜃 .
ℎ𝜃
0.6
0.5
0.4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
3
4
5
Ratio of the length of pole
F 14: e width of ℎ𝑥𝑥 platform varies with the ratio of the
length.
F 11: ℎ𝑥𝑥 varies with ℎ𝜃𝜃 when the �tness is the maximum.
Correlation coefficient
2
No. 𝑚𝑚1 (kg)
0
Correlation coefficient
1
Actual value
Predicted value
F 10: e maximum �tness varies with ℎ𝜃𝜃 .
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.65
1
5
6
7
8
1
2
3
4
ℎ𝑥
0.7
0.7
0.8
0.9
1
F 13: e correlation coefficient between �tness and ℎ𝜃𝜃 varies
with ℎ𝑥𝑥 .
1
2
3
4
5
6
7
8
0.0149
0.0216
0.0284
0.0378
0.0493
0.0621
0.0773
0.0966
𝐿𝐿1 (m)
0.1
0.149
0.199
0.266
0.354
0.4436
0.553
0.691
e correlation
coefficient between
the maximum
�tness and ℎ𝑥𝑥
0.3801
0.3956
0.4104
0.4261
0.4285
0.4294
0.4308
0.4327
e correlation
coefficient between
the maximum
�tness and ℎ𝜃𝜃
0.9602
0.9503
0.9537
0.9604
0.9513
0.9314
0.9125
0.8895
coefficient between the maximum �tness and ℎ𝑥𝑥 is
small, but the one between the maximum �tness and
ℎ𝜃𝜃 is big. Table 6 shows that the longer the pole, the
smaller the optimum ℎ𝑥𝑥 , and the optimum ℎ𝜃𝜃 is in the
range 0.8 to 0.9 with little difference.
From the third conclusion, we can know that the width of
ℎ𝑥𝑥 platform varies with the length of pole. at is, the longer
the pole, the smaller the width. For poles of different lengths,
the corresponding ℎ𝑥𝑥 platforms are shown in Table 8. And the
relation between the width of ℎ𝑥𝑥 platform and the length of
pole and the one between the starting value of ℎ𝑥𝑥 platform
and the length of pole can be obtained through �tting method
according to the experimental results shown in Table 8.
Firstly, suppose that the smallest effective length of pole,
that is, 0.1 m, is 1 unit. en, the ratios of the other lengths to
the smallest effective length of pole are shown in Table 8. e
relation between the width of ℎ𝑥𝑥 platform and the ratio of the
length of pole is depicted by means of the linear �t, which is
as shown in (50) and Figure 14. Consider
𝑦𝑦 𝑦 𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝑦 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝑦 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦2 .
(50)
Discrete Dynamics in Nature and Society
No.
T 8: e corresponding ℎ𝑥𝑥 platforms for poles of different lengths.
𝐿𝐿1 (m)
1
2
3
4
5
6
7
8
13
e starting value
of ℎ𝑥𝑥 platform
Ratio of the lengths of poles
0.1
0.149
0.199
0.266
0.354
0.4436
0.553
0.691
1
1.49
1.99
2.66
3.54
4.436
5.53
6.91
0.175
0.22
0.235
0.25
0.26
0.27
0.275
0.285
Starting value of ℎ𝑥 platform
0.3
0.28
0.26
0.24
0.22
0.2
0.18
0.16
1
2
3
4
5
Ratio of the length of pole
6
7
Actual value
Predicted value
F 15: e starting value of ℎ𝑥𝑥 platform varies with the ratio of
the length.
Similarly, the relation between the starting value of ℎ𝑥𝑥
platform and the ratio of the length of pole is depicted by
means of the linear �t, which is as shown in (51) and Figure
15. Consider
𝑦𝑦 𝑦 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝑦 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝑦 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦2 +0.0015301𝑥𝑥3 .
(51)
Hence, for the given physical system, when the length
of pole takes different values in the previous controllable
interval, we can calculate the corresponding interval of ℎ𝑥𝑥
platform by (50) and (51). en, ℎ𝑥𝑥 can be any of the interval
and ℎ𝜃𝜃 should be any of the interval [0.8, 0.9]. erefore, it
does not need to optimize the control parameters again, and
we can realize the effective control for the physical system
with new length of pole.
4. Conclusion
Flexible logic method uses universal combinatorial operation
model to describe the logic relation between 𝐸𝐸 and EC, which
are the fuzzy input variables of normal two-dimensional
fuzzy controller. Universal combinatorial operation model
is not a single �xed operator, but a continuous cluster of
e �nishing value
of ℎ𝑥𝑥 platform
0.98
0.98
0.98
0.975
0.955
0.94
0.94
0.94
e width
of ℎ𝑥𝑥 platform
0.805
0.76
0.745
0.725
0.695
0.67
0.665
0.655
combinatorial operators determined by the general correlation coefficient ℎ between propositions. In practical control
application, according to the general correlation between
propositions, we can take the corresponding one from the
cluster to realize effective control for complex system.
However, in practical control application, how to determine the general correlation coefficient ℎ of �exible logic
control model is a problem for further studies. First, the
conventional universal combinatorial operation model has
been limited in the interval [0, 1]. Consequently, this paper
studies a kind of universal combinatorial operation model
based on any interval [𝑎𝑎𝑎 𝑎𝑎𝑎. And some important theorems
are given and proved, which provide a foundation for the
�exible logic control method. For dealing reasonably with the
complex relations of every factor in complex system, a kind
of universal combinatorial operation model with unequal
weights is put forward. en, this paper has carried out
the parametric analysis of �exible logic control model. And
some research results have been given, which have important
directive to determine the values of the general correlation
coefficients in practical control application.
Acknowledgments
is work is supported by the Beijing Municipal Natural
Science Foundation (nos. 4113069 and 4122007), the Major
State Basic Research Development Program of China (no.
2007CB311100), and the Beijing Municipal Education Commission Foundation (no. 007000546311501).
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